Error estimates for nonconforming and discontinuousdiscretizations of nonsmooth problems via convex duality
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Soren BartelsUniversity of Freiburg, Germany
Joint seminar UMD College Park & Baltimore, George Mason U, U Delaware
Sayas Numerics Seminar, September 7, 2021
Joint work with Robert Tovey (INRIA), Zhangxian Wang, Friedrich Wassmer
Outline
1 TV Model Problem
2 Convexity and Nonstandard FE
3 Nonsmooth Examples
4 Generalization to dG
5 Mesh Grading and Adaptivity
6 Summary
h
L2 projectionx 7→ sign(x)
−G
F
−G(w)
F (w)
−F∗(s)
G∗(s)
S. Bartels Discontinuous methods for nonsmooth minimization 2/34
TV Model Problem
Reduced rates
Example: Image processing via TV minimization [Rudin, Osher & Fatemi ’92]
I (u) = |Du|(Ω) +α
2‖u − g‖2
I nondifferentiable convex problem with discontinuous solutionsI formal Euler-Lagrange equations − div ∇u
|∇u| + α(u − g) = 0
Experiment:
P1-FEh ∼ 2−5
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Accuracy: Error estimates for L2 error if u ∈ L∞(Ω) ∩ BV (Ω)
I O(h1/4) for P1 FE [Wang & Lucier ’11, B. ’12, B., Nochetto & Salgado ’14]
I O(h1/2) for anisotropic TV [B., Nochetto & Salgado ’15]
I O(h1/2) for P0/CR FE [Chambolle & Pock ’19+, B. ’20+]
Note: Rate O(h1/2) optimal for generic BV functions
h
L2 projectionx 7→ sign(x)
S. Bartels Discontinuous methods for nonsmooth minimization 4/34
General approach
Variational problem: With convex functionals φ, ψ
I (u) =
∫Ω
φ(∇u) dx +
∫Ω
ψ(x , u)dx
I formal Euler-Lagrange equations: − div φ′(∇u) + ψ′(·, u) = 0
Coercivity: Optimality of u
δI (u, uh) ≤ I (uh)− I (u)
Error estimate: By minimality of uh for every vh
δI (u, uh) ≤ I (vh)− I (u)u uh
δI (u, uh)
I
Hence: Suitable interpolant vh = Jhu leads to error estimate
I control of inconsistent Ih ≈ I
I optimality and regularity
I use of duality I (u) ≥ D(z)
S. Bartels Discontinuous methods for nonsmooth minimization 5/34
Limitations of P1-FE
Model: ROF minimization in BV (Ω), discretization with P1-FE
α
2‖u − uh‖2 ≤ I (vh)− I (u)
Interpolant: With regularization uε set vh = Ihuε
‖u − Ihuε‖L1 ≤ c(h2ε−1 + ε)|Du|(Ω),
‖∇Ihuε‖L1 ≤ (1 + chε−1 + cε)|Du|(Ω)
h
ε
uuεIhuε
Continuity: With binomial formula and ε = h1/2
α
2‖u − uh‖2 ≤ |Dvh|(Ω)− |Du|(Ω) +
α
2
∫Ω
(vh − u)(vh + u − 2g) dx
≤ chε−1 + c(h2ε−1 + ε) = O(h1/2)
Optimality via TVD: If Ih : W 1,1(Ω)→ S1(Th) with ‖∇Ihv‖L1 ≤ ‖∇v‖L1 then
α
2‖u − uh‖2 = O(h)
OK: Anisotropic `1 version of BV and structured grids [B., Nochetto & Salgado ’15]
S. Bartels Discontinuous methods for nonsmooth minimization 6/34
Convexity and Nonstandard FE
Fenchel conjugates
Conjugate: For convex φ : R` → R define convex conjugate
φ∗(s) = supr∈R`
r · s − φ(r)
i.e., Young’s ineq r · s ≤ φ(r) + φ∗(s), equality for s = φ′(r)
Duality: Use φ∗∗ = φ, integrate by parts, exchange extrema
infuI (u) = inf
u
∫Ω
=φ(∇u)︷ ︸︸ ︷supz∇u · z − φ∗(z) dx +
∫Ω
ψ(x , u) dx
≥ supz
infu−∫
Ω
φ∗(z) dx −∫
Ω
u div z − ψ(x , u)dx
= supz−∫
Ω
φ∗(z) dx −∫
Ω
ψ∗(x , div z) dx = supz
D(z)
r
φ(r)
−φ∗(s)
r · s
φ
−φ∗(s) = infr Ls,r (0)
infw F (w) + G(w)
= sups −F∗(s) − G∗(s)
−G
F
−G(w)
F (w)
−F∗(s)
G∗(s)
Strong duality: If saddle (u, z) exists then equality I (u) = D(z) holds
I important implicit information in dual solution z = φ′(∇u)
S. Bartels Discontinuous methods for nonsmooth minimization 8/34
Crouzeix–Raviart FE
Motivation: Continuous P1 vector fields too stiff for divergence, e.g. in Stokes,
div : S1(Th)d → L0(Th) NOT surjective
CR-FEM: S1,cr (Th) continuous (only) at side midpoints [Crouzeix–Raviart ’73]
Quasi-interpolant: Averages over element sides
Jh : H1(Ω)→ S1,cr (Th), Jhv(xS) = |S |−1
∫S
v ds
Projection property: Elementwise gradient given by projection onto constants
∇hJhv = Πh∇v
Jensen’s inequality: For convex function φ∫Ω
φ(∇hJhv)dx ≤∫
Ω
φ(∇v) dx
S. Bartels Discontinuous methods for nonsmooth minimization 9/34
Raviart–Thomas FE
Motivation: Divergence operator on L2 vector fields (instead H1)
RT-FEM: RT 0(Th) affine vector fields with weak divergence [Raviart–Thomas ’77]
qh|T (x) = ax + b
a ∈ R, b ∈ Rd
Idea: Constant normal components on hyperplanes
qh(x) · nS = c ∀x ∈ S nS
Sx
x ′
Quasi-interpolant: Averages of normal components over sides
Lh : H(div; Ω)→RT 0(Th), Lhq(xS) · nS = |S |−1
∫S
q · nS ds
Projection property: Surjectivity of div : RT 0(Th)→ L0(Th) via
divLhq = Πh div q
S. Bartels Discontinuous methods for nonsmooth minimization 10/34
CR-RT duality
Properties of CR and RT: On element sides S ∈ Sh with normal nS
I jumps JvhK of CR functions over sides have vanishing mean
I components qh · nS for RT fields constant and continuous
Integration by parts: For vh ∈ S1,crD (Th) and qh ∈ RT 0
N(Th)∫Ω
vh div qh dx = −∑T∈Th
∫T
∇vh · qh dx +∑S∈Sh
∫S
JvhKqh · nS ds︸ ︷︷ ︸=0
= −∫
Ω
∇hvh · qh dx
Projection onto constants: L2 projection Πh : L2(Ω;R`)→ L0(Th)`
Duality connection: Within L0(Th)`∫Ω
Πhvh div qh dx = −∫
Ω
∇vh · Πhqh dx
I projection onto constants important to apply Fenchel’s inequality
S. Bartels Discontinuous methods for nonsmooth minimization 11/34
Discrete Fenchel duality
Imitate arguments: For uh ∈ S1,crD (Th) and zh ∈ RT 0
N(Th)
Ih(uh) =
∫Ω
φ(∇huh) + ψh(Πhuh) dx
≥∫
Ω
−φ∗(Πhzh) + Πhzh · ∇huh + ψh(Πhuh) dx
=
∫Ω
−φ∗(Πhzh)− div zh Πhuh + ψh(Πhuh)dx
≥∫
Ω
−φ∗(Πhzh)− ψ∗h (div zh)dx
= Dh(zh)
Weak duality: Discrete functionals Ih and Dh satisfy Ih(uh) ≥ Dh(zh)
Strong duality: If φ, ψh ∈ C 1 and uh minimal for Ih then
zh = Dφ(∇huh) + Dψh(·,Πhzh)d−1(x − xT ) =⇒ Ih(uh) = Dh(zh)
Related: zh = Dφ(∇huh) + d−1(x − xT )Πhf [Marini ’85, Carstensen & Liu ’15]
S. Bartels Discontinuous methods for nonsmooth minimization 12/34
Error analysis via duality
Nonlinear Dirichlet problem: With fh = Πhf consider
Ih(uh) =
∫Ω
φ(∇huh) dx −∫
Ω
fh Πhuh dx ,
Dh(zh) = −∫
Ω
φ∗(Πhzh) dx − I−fh(div zh)
A priori analysis: Control discrete error δIh (uh,Jhu) via
δIh ≤ Ih(Jhu)− Ih(uh) ≤ Ih(Jhu)− Dh(Lhz)
≤∫
Ω
φ(∇u)− Lhzh · ∇hJhu + φ∗(ΠhLhz) dx
≤∫
Ω
−φ∗(z) + (z − ΠhLhz) · ∇u + φ∗(ΠhLhz) dx
≤∫
Ω
(Dφ∗(z)− Dφ∗(ΠhLhz)
)· (z − ΠhLhz) dx
= O(h2)
I discrete coercivity & duality
I Jensen’s inequality,divLhz = −fh
I strong duality I (u) = D(z),
div z = −f , ∇hJhu = Πh∇u
I convexity, ∇u = Dφ∗(z)
I interpolation estimates,
if, e.g., φ∗ Lipschitz
S. Bartels Discontinuous methods for nonsmooth minimization 13/34
General error bound
Generalization: Convex variational problems on W 1,pD (Ω)
Theorem ([B. ’20+]). If z ∈W qN(div; Ω) ∩W 1,1(Ω;Rd)
δIh (uh,Jhu) ≤∫
Ω
(Dφ∗(z)− Dφ∗(ΠhLhz)
)· (z − ΠhLhz) dx
+
∫Ω
(Dψ(u)− Dψh(ΠhJhu)
)· (u − ΠhJhu) dx
+
∫Ω
ψh(u)− ψ(u)dx +
∫Ω
ψ∗h (Πh div z)− ψ∗(div z)dx .
Interpretation: CR discretization error controlled by RT interpolation error
I inconsistency ∇h ≈ ∇ and Ih ≈ I estimated via duality
I low order terms reduce to (γ/2)‖u − ΠhJhu‖2 for quadratic ψ
I avoids use of Strang lemmas; guarantee feasibility of interpolants
I use of quadrature important and computationally efficient
S. Bartels Discontinuous methods for nonsmooth minimization 14/34
Hidden structure
Orthogonality relation: Within (L0)d ⊂ L2(Ω;Rd) [B. & Wang ’20+](ΠhRT 0
N
)⊥= ∇h
(ker Πh|S1,cr
D
)⇐⇒: A⊥ = B
Proof: (i) If Πhvh = 0 then for all yh ∈ RT 0N
(∇hvh,Πhyh) = −(Πhvh, div yh) = 0 =⇒ A ⊂ B⊥
(ii) Let yh ∈ B⊥. Choose rh ∈ Zh = (ker Πh|S1,crD
)⊥ ⊂ S1,crD with
(Πhrh,Πhvh) = (yh,∇hvh) ∀vh ∈ Zh.
True for all vh ∈ S1,crD . Let zh ∈ RT 0
N with − div zh = Πhrh
(yh − zh,∇hvh) = (Πhrh,Πhvh) + (div zh, vh) = 0 ∀vh ∈ S1,crD
Define yh|T = yh|T + div zh|T (x − xT )/d so that
(yh − zh,∇hvh) = (yh − zh,∇hvh) = 0 ∀vh ∈ S1,crD
Implies yh − zh ∈ RT 0N . Since Πh yh = yh deduce yh ∈ A and B⊥ ⊂ A.
Remarks: Identity uses inf-sup condition and has consequences
I strong discrete duality relation
I dual variant div ker Πh|RT 0N
=(ΠhS1,cr
D
)⊥=⇒ ΠhS1,cr
D = L0 if ΓD 6= ∂Ω
I alternative proof uses discrete Poincare lemma [Chambolle & Pock ’19+]
S. Bartels Discontinuous methods for nonsmooth minimization 15/34
Nonsmooth Examples
Example: Obstacle problem
Functionals: For f ∈ L2(Ω) impose u ≥ 0 via indicator functionals I+
I (u) =1
2
∫Ω
|∇u|2 dx −∫
Ω
fu dx + I+(u),
D(z) = −1
2
∫Ω
|z |2 dx − I−(div z + f )
f
u
Low order terms: Pointwise convex conjugation
ψ(x , s) = −f (x)s + I+(s) =⇒ ψ∗(x , t) = I−(t + f (x))
Complementarity: z = ∇u and div z + f = 0 if u > 0
Discretization: Use fh = Πhf and impose constraint at midpoints of elements
Ih(uh) =1
2
∫Ω
|∇huh|2 dx −∫
Ω
fhuh dx + I+(Πhuh),
Dh(zh) = −1
2
∫Ω
|Πhzh|2 dx − I−(div zh + fh)
S. Bartels Discontinuous methods for nonsmooth minimization 17/34
Example: Obstacle problem (cont’d)
Proposition ([B. ’20+]). If z = ∇u ∈ H1(Ω;Rd) then
‖∇h(uh − u)‖ ≤ ch(‖D2u‖+ ‖f + div z‖
).
Proof: Since Jhu and Lhz admissible with µ = f + div z
δ2h ≤
∫Ω
(f + div z)(u − ΠhJhu) dx +1
2
∫Ω
|z − ΠhLhz |2 dx
=
∫Ω
µ(Jhu − ΠhJhu) dx +
∫Ω
µ(u − Jhu) dx +1
2
∫Ω
|z − ΠhLhz |2 dx .
For element contact set CT = x ∈ T : u(x) = 0
µ|T\CT = 0, ∇u|CT = 0.
With Jhu(x) = Jhu(xT ) +∇hJhu|T · (x − xT ) for critical term A
A|T =
∫CTµ∇h(Jhu − u) · (x − xT ) dx ≤ hT‖µ‖L2(T )‖∇h(Jhu − u)‖L2(T ).
I note constant-free intermediate estimate
S. Bartels Discontinuous methods for nonsmooth minimization 18/34
Example: TV minimization
Primal problem: For noisy image g ∈ L∞(Ω) find u via minimizing
I (u) = |Du|(Ω) +α
2‖u − g‖2
Total variation: Extends ‖∇u‖L1(Ω) via operator norm of Du on C(Ω)
|Du|(Ω) = sup−∫
Ω
u div z dx : z ∈ C∞c (Ω;Rd), |z |`2 ≤ 1 in Ω
Dual problem: By exchanging extrema u = g + α−1 div z
D(z) = − 1
2α‖ div z + αg‖2 +
α
2‖g‖2 − IK1(0)(z)
Optimality: z ∈ ∂|∇u| and strong duality I (u) = D(z) [Hintermuller & Kunisch ’04]
Discretization: With gh = Πhg consider inconsistent Ih and Dh
Ih(uh) =
∫Ω
|∇huh|dx +1
2‖Πhuh − gh‖2,
Dh(zh) = − 1
2α‖ div zh + αgh‖2 +
α
2‖gh‖2 − IK1(0)(Πhzh)
S. Bartels Discontinuous methods for nonsmooth minimization 19/34
Example: TV minimization (cont’d)
Proposition ([Chambolle & Pock ’19+, B. ’20+]). If z ∈W 1,∞(Ω;Rd) then
‖u − Πhuh‖ ≤ ch1/2(‖u‖L∞ |Du|(Ω) + ‖g‖‖∇z‖L∞‖ div z‖)1/2
.
Proof: For admissible uh and zh
1
2‖Πh(uh − uh)‖2 ≤ Ih(uh)− Ih(uh) ≤ Ih(uh)− Dh(zh)
For uh = limε→0 Jhuε have TVD and approximation
‖∇huh‖L1 ≤ |Du|(Ω), ‖uh − u‖L1 ≤ ch|Du|(Ω),
and hence
Ih(uh) ≤ I (u) + ch − 1
2‖g − gh‖2.
h
u
uh
For zh = γ−1h Lhz with γh = ‖Lhz‖L∞ have
Dh(zh) ≥ D(z)− ch − 1
2‖g − gh‖2.
Strong duality I (u) = D(z) gives estimate.
I note canceling low order contributions; no regularity conditions on g
S. Bartels Discontinuous methods for nonsmooth minimization 20/34
Example: TV minimization (cont’d)
Setup: For Dirichlet BC, g = χBr (0) have
u = max0, 1− d/(αr)χBr (0), z ∈W 1,∞(Ω;Rd)
I choose r = 1/2, d = 2, Ω = (−1, 1)2, and α = 10I iterative solution via semi-implicit gradient descent [B., Diening & Nochetto ’18]
P1 solution up1h and Πhu
p1h
CR solution ucrh and Πhu
crh
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S. Bartels Discontinuous methods for nonsmooth minimization 21/34
Irregular dual solution
Single disc phantom [Chambolle et al. ’10]: For B = B1(0) ⊂⊂ Ω and g = χB ,Dirichlet BC, ROF minimizer u = cαg with Lipschitz continuous dual
z(x) = −c ′α
x , |x | ≤ 1
x/|x |2, |x | ≥ 1
Optimality: Dual solutions satisfy z ∈ ∂|Du| and div z = −α(u − g)
Touching discs [B., Tovey & Wassmer ’21+]:For symmetric Ω, g = χB+ − χB− , Dirichlet BC,have u = c ′1,αg and z = ∓ν on ∂B± and
|x+ − x−| ≈ ϕ2, |z(x+)− z(x−)| ≈ 2ϕ
=⇒ z not Lipschitz continuous
ϕϕB+rB−
r
x1
x−
z(x−)x+
z(x+)
x2
Ω
No reduction ofCR convergencerate:
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S. Bartels Discontinuous methods for nonsmooth minimization 22/34
Generalization to dG
Discontinuous Galerkin method
Jumps and averages: On inner sides S ∈ Sh with normal nS
JvhK(x) = limε→0
(vh(x − εnS)− vh(x + εnS)
),
vh(x) = limε→0
1
2
(vh(x − εnS) + vh(x + εnS)
)JvhK|S vh|S
I midpoint evaluation JvhKh|S = JvhK(xS) and vhh|S = vh(xS)
Broken CR and RT spaces: Omit all continuity requirements
S1,dg (Th) = L1(Th), RT 0,dg (Th) = L0(Th)d + (id−xT )L0(Th)
Integration by parts: With JvhyhK = JvhKyh+ vhJyhK∫Ω
vh divh yh dx +
∫Ω
∇hvh · yh dx
=
∫Sh\ΓN
JvhKhyh · nS ds +
∫Sh\ΓD
vhhJyh · nSKds
I reveals duality of jumps and averages
S. Bartels Discontinuous methods for nonsmooth minimization 24/34
dG-Discretized problems
Stabilizing and regularizing terms: For uh ∈ S1,dg (Th) and zh ∈ RT 0,dg (Th)
Jh(uh) =1
r‖α−1S JuhKh‖rLr (Sh\ΓN ) +
1
s‖βSuhh‖sLs (Sh\ΓD ),
Kh(zh) =1
r ′‖αSzh · nS‖r
′
Lr′
(Sh\ΓN )+
1
s ′‖β−1S Jzh · nSK‖s
′
Ls′
(Sh\ΓD )
Proposition ([B. ’20+]). Ih(uh) ≥ Dh(zh) for primal and dual dG functionals
Ih(uh) =
∫Ω
φ(∇huh) + ψh(x ,Πhuh) dx + Jh(uh),
Dh(zh) = −∫
Ω
φ∗(Πhzh) + ψ∗h (x , divh zh)dx − Kh(zh).
If r = s = 2 and βS = 0 and φ, ψ ∈ C 1 then strong duality applies.
I use of quadrature in jumps and averages essential
S. Bartels Discontinuous methods for nonsmooth minimization 25/34
Error estimate
Nonlinear Dirichlet problems: Consider low order term ψ(x , s) = −f (x)s
Proposition ([B. ’20+]). If z ∈W 1,1(Ω;Rd)
δIh (uh,Jhu) ≤∫
Ω
(Dφ∗(z)− Dφ∗(ΠhLhz)
)· (z − ΠhLhz) dx
+ Jh(Jhu) + Kh(Lhz),
where
Jh(Jhu) ≤ cT ‖h−1S βs
S‖L∞(S)‖Jhu‖sLs (Ω),
Kh(Lhz) ≤ cT ‖h−1S αr′
S ‖L∞(S)‖Lhz‖r′
Lr′
(Ω).
Examples: Strength of jump termsI Poisson: r = 2 and αS = h3/2 (sharp)I TV: r = 1 and αS ≤ 1 or r = 2 and αS = hp, p ≥ 2
Consistency: Discretize then optimizeI note disappearing jump terms for CR and RT comparison functionsI lack of consistency terms for elliptic problems requires overpenalization
S. Bartels Discontinuous methods for nonsmooth minimization 26/34
Numerical experiment
Regularized functional: With |a|ε = (|a|2 + ε2)1/2
Ih,ε(uh) =
∫Ω
|∇huh|ε dx +α
2‖Πhuh − gh‖2 +
c−rα
r
∫Sh−γrS |JuhK|rε ds
I uniform estimate 0 ≤ |a|ε − |a| ≤ ε justifies ε = h
Linear: r = 1, γ = 1, cα = 1
Quadratic: r = 2, γ = 1, cα = 1
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S. Bartels Discontinuous methods for nonsmooth minimization 27/34
Mesh Grading and Adaptivity
Approximation of 1D jumps
L2 Best approximation: For u = sign +v , v ∈ H1(a, b)
c1h1/20 ≤ inf
uh∈S1(Th)‖u − uh‖ ≤ c2
(h
1/20 + hmax‖v ′‖
)Idea: Use β-graded mesh with hmin ∼ hβmax
1D ROF model: If u ∈W 2,1((a, b) \ Ju)
α
2‖u − uh‖2 ≤ I (Ihu)− I (u)
= ‖(Ihu)′‖L1(a,b) − |Du|(a, b)︸ ︷︷ ︸≤0 (TVD property of Ih)
+α
2
∫ b
a
(Ihu − g)2 − (u − g)2 dx︸ ︷︷ ︸≤4‖u−Ihu‖L1‖g‖L∞
≤ cα(hmin|Du|(a, b)
+ h2max‖u′′‖L1((a,b)\Ju)
)‖g‖L∞
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S. Bartels Discontinuous methods for nonsmooth minimization 29/34
2D Mesh grading
Graded triangulations: Shape regularity andconformity limit grading strength in Rd , d > 1
CR discretization: Use discrete TVD property and discrete duality as aboveα
2‖Πh(uh − uh)‖2 ≤ 2α‖Πhuh − u‖L1(Ω)‖g‖L∞(Ω) + (1− γ−1
h )‖g‖‖ div z‖
Difficulty: Local error O(hT ) even in regular regions, γh ≥ maxT∈Th |JRT z(xT )|
Proposition ([B., Tovey & Wassmer ’21+]). If ROF sol.u ∈ BV (Ω) p/w constant, dual z ∈W 1,∞(Ω;Rd) s.t.
|z(x)| ≤ 1− `dJu (x)
and triangulations quadratically graded
hT ≤ c
hdJu (T )1/2 if dJu (T ) ≥ h2,
h2 otherwise
then ‖Πh(u − uh)‖ ≤ h| log h|M(α, u, z , g).
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S. Bartels Discontinuous methods for nonsmooth minimization 30/34
A posteriori error estimate
Continuous coercivity: For conforming approximation uch ∈W 1,p(Ω)
δI (u, uch) ≤ I (uc
h)− I (u) ≤ I (uch)− D(zh)
=
∫Ω
φ(∇uch) + ψ(uc
h) + φ∗(zh) + ψ∗(div zh)dx
Poisson problem: Assume fh = f and use div zh = −f
1
2‖∇(u − uc
h)‖2 ≤ 1
2
∫Ω
|∇uch |2 + |zh|2 + div z uc
h dx =1
2‖∇uc
h − zh‖2
TV minimization: Direct calculations and integration by parts for |zh| ≤ 1
α
2‖u − uc
h‖2 ≤∫
Ω
|∇uch |+
α
2(uc
h − g)2 +1
2α(div z + αg)2 − α
2g 2 dx
=
∫Ω
(|∇uc
h | − ∇uch · zh
)+
1
2α(div zh + α(uc
h − g))2 dx
Interpretation: Residual type estimators, uch and zh only admissible
I flux equilibration, hypercircle method: [Repin ’00, Braess ’07, B. ’15]
I nearly optimal zh via postprocessing primal CR approximation uh
S. Bartels Discontinuous methods for nonsmooth minimization 31/34
Adaptive experiment
ROF estimator: Use localized error bound to refine mesh [B. & Milicevic ’19]
e2h ≤ η2
h =∑T∈Th
∫T
(|∇uc
h | − ∇uch · zh
)+
1
2α(div zh + α(uc
h − g))2 dx
100
101
102
103
104
105
10-2
10-1
100
cN-0.29
cN-0.29
cN-0.38
I use of H(div; Ω)-conforming dual variable avoids oscillationsI open: construction of admissible, nearly optimal z (h)
S. Bartels Discontinuous methods for nonsmooth minimization 32/34
Summary
Summary
I Optimal error estimates via convex duality
I Nonsmooth problems, only first order systems
I Dual problem provides right regularity condition
I A posteriori error estimates via primal-dual gap
I Solve dual problem via generalized Marini formula
I Future aspects/open problems:
B general regularity results, e.g., dual form TV
B higher order nonstandard finite elements
B optimal iterative solution
I More information
http://aam.uni-freiburg.de/bartels
Thank you !
S. Bartels Discontinuous methods for nonsmooth minimization 34/34